In our scripts I found the Professor denoting the complex Fourier series slightly differently
First of the series for a function $f(x) \in V\: $ he defines as:
$\displaystyle f = \lim_{n \to \infty} \sum_{k = 1}^n\langle e_k,f\rangle\,e_k$ where $e_k = \dfrac{1}{\sqrt{2\,\pi}}\,e^{\textstyle i\,k\,x}\quad \text{thus} \quad f =\displaystyle \dfrac{1}{2\,\pi}\int_{-\pi}^\pi e^{\textstyle -i\,k\,x}f(x)\,\mathrm{dx}\,e^{\textstyle i\,k\,x}$
Here I wonder: did he start the sum at $k = 1$ instead of $-\infty$ by mistake or is it safe to say?
Secondly he continues defining:
$ \displaystyle S_n(x) = \sum_{k =-n}^{n} c_k\,e^{\textstyle i\,k\,x}$ where $\displaystyle c_k = \dfrac{1}{2\,\pi}\int_{-\pi}^\pi e^{\textstyle -i\,k\,x}f(x)\,\mathrm{dx} \quad \text{as well as}$
$\displaystyle \sigma_n(x) = \dfrac{1}{n}\,\left(S_0+ \ldots+ S_{n-1}\right) \quad \text{and conludes}$
$\textbf{Conclusion:}\:\displaystyle \lim_{n \to \infty} \sigma_n = f(x)$
What? In my view a Fourier series is to be defined as: $ f(x) = \displaystyle \lim_{n\to \infty}S_n$
So why all those different notations?